Use of formulae for the sums of integers, squares and cubes

Understanding the Sums of Integers, Squares and Cubes

The sum of the first (n) positive integers can be represented as (\frac{n(n+1)}{2}).
The sum of the squares of the first (n) positive integers can be represented by (\frac{n(n+1)(2n+1)}{6}).
The sum of the cubes of the first (n) positive integers can be represented by (\left[\frac{n(n+1)}{2}\right]^2).

Using the Formulae

These formulae can be used to compute the sums easily without the need to calculate each term individually.
They are especially useful in series and sequences for computing the sum of the first (n) terms of arithmetic series, square series and cubic series.

Proofs of the Formulae

The proofs for these formulae rely on concepts such as mathematical induction and series manipulation.
For example, the formula for the sum of cubes can be proved using the identity (k^3 - (k-1)^3 = 3k^2 - 3k + 1) and mathematical induction.

Applications of the Formulae

These formulae have practical applications in many areas of mathematics including calculus, number theory and geometry.
For example, in number theory, these formulae are used to find perfect numbers, triangular numbers, etc.

Remember, practice and understanding of these formulae is important for successful problem solving in further pure mathematics. These formulae are fundamental tools you will use when working with sums of sequences and series.